1. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Valence bond solid order through spin-lattice coupling Jung Hoon Han & Chenglong Jia Sung Kyun Kwan U. 成均館大 Ref: Jia(Kim) 2 Han, PRB (2005); Jia & Han, cond-mat/0505573

2. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Lattice-Coupled Antiferromagnetic Spin Model Spin and lattice are coupled through magneto-striction effect (Pytte, PRB 1974) i j uiui ujuj R ij Lattice displacement related to local spin-spin correlation by

3. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Application to Frustrated Lattices – Triangular, Kagome NO LATTICE DEFORMATION!

4. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Triangular/Kagome lattices – meanfield theory Ground state of Heisenberg model gives 120 degrees for neighboring spins for triangular/Kagome According to linearized spin-lattice theory local displacement is zero. Classical degeneracy of Kagome is not lifted.

5. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU 3D Frustrated Lattice - Pyrochlore Tetrahedron as a building block Ground state condition for each tetrahedron Macroscopic classical ground state degeneracy in pristine Heisenberg model

6. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Lifting of degeneracy through spin-lattice coupling For pyrochlore lattice, there are infinitely many ways to deform the lattice through linearized spin-lattice coupling Some clues to the deformation pattern must come from experiment

7. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Experiments on Pyrochlore – ZnCr2O4 Below Tc S.H.Lee et al. PRL, 2000 : Spins on Cr 3+ (S=3/2) order antiferromagnetically at as first-order transition, acccompanied by cubic-to-tetragonal distortion.

8. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Theory of spin-Peierls transition by Tschernyshyov, Moessner, and Sondhi TMS, PRL, 2002 TMS, PRB, 2002 Elongation (contraction) of a tetrahedron along an axis And collinear antiferromagnetic spins is the ground state

9. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Experiments on Pyrochlore – ZnCr2O4 Above Tc S.H.Lee et al. Nature, 2002 : Neutron scattering of paramagnetic state at T=15K Structure factor consistent with hexagon spin cluster (spin-loop director) Conclude: Neutrons scatter off a hexagon cluster of spins, rather individual, fluctuating spins (very unusual paramagnetic state!)

10. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Lifting of degeneracy through spin-lattice coupling Tetrahedron Unit (Tschernyshyov et al. PRL 2002) Hexagon Unit (Jia et al. PRB 2005)

11. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Interpreting Experiments as Spin-Lattice Coupling Invoking spin-lattice coupling, each independent hexagon shrinks, taking advantage of finite lattice stiffness and lowering exchange energy Directors of nearby hexagons interact via

12. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Spins within a hexagon are collinear Spins of nearby hexagons are orthogonal Director-Director Interaction

13. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Hexagon Organization of Spins in Pyrochlore Hexagons form a super-lattice with each color representing a director orientation All nearby directors are orthogonal (3-states Potts)

14. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Energy Comparison In comparing energies for both distortion modes, Tetrahedron-based distortion has a lower energy, but hexagon shrinkage is a viable, metastable state

15. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU A picture of paramagnetic state in ZnCr 2 O 4 Spin-lattice interaction leads to enhanced singlet (collinear antiferromagnet) tendency within a hexagon Residual spin-lattice interaction leads to orthogonality of nearby directors (3-states Potts model) At finite temperature, thermal fluctuations smear out the inter-hexagon interaction, spin-spin correlation remains confined to a single hexagon Further lowering temperature might lead to condensation of spin-loop directors, but it appears that a tetragonal distortion pre-empts this possibility in ZnCr 2 O 4

16. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Spin-Lattice Coupling in YMnO 3 Mn 3+ (S=2) spins form triangular network in MnO 2 plane Katsufuji et al. PRB (2001)

17. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Experiments on triangular S=2 AFM – YMnO 3 Expansion of Mn triangles (red) at the onset of magnetic order But simple spin-lattice theory won’t work…. S Lee et al. PRB (2005)

18. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU New order parameter ? Possible order parameters of spin: (1) spin density wave – magnetism (2) valence-bond – VBS or VBL(RVB)

19. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU 1D Chain (S=1) 2D square (S=2) 2D hexagonal (S=3/2) Examples of VBS (AKLT states)

20. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU IDEA: Spin-Lattice Coupling-induced VBS ? Magnetic order doesn’t work, but valence-bond order might. PARTIAL VBS ORDER At most 4 VB’s can form for coordination = 6 Without SLC, VB’s will resonante, but with SLC, VBS patterns can be formed.

21. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Calculations Take spin-lattice Hamiltonian and use Schwinger boson mean-field theory (Jia & Han, cond-mat/0505573) Bond order Bose condensation Spin singlet LRO

22. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Results Energy corresponding to a particular VBS pattern below is lower than that of uniform state for sufficiently large spin-lattice coupling strength (Not spin Jahn-Teller type)

23. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Results This state is also Bose-condensed, so LRO co-exists with VBS Elementary excitations: (1) Spin waves for |LRO> (2) Triplet wave for |PVBS>

24. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU A New Excitation We calculate excitation spectra for the VBS within SMA VBS Spin waveComparison

25. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Interpretation (1) A new spectrum is gapped, has a minimum at (4pi/3a,0), same as spin wave minimum (2) Dispersion is extremely flat! Reminiscent of the flat band in kagome AFM.

26. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Relation to Experiments? – YMnO 3 Tc=70K Quasi-elastic peak at high T, into Neel region (Takagi: fluctuating XY spins but 40K seems too far from Tc) Any sign of flat band? (must extract spin waves first) Park et al. PRB (2003)

27. KIAS Emergence Workshop 2005 Manybody Physics Group SKKU Summary A naïve inclusion of magneto-striction leads to a variety of interesting phenomena (1) Classical spin degeneracy of pyrochlore lattice can be lifted through magneto-striction in a number of interesting ways; tetrahedron-based or hexagon-based (2) Formation of valence-bond-solid is facilitated by the spin-lattice coupling and may lead to an interesting excitation spectrum